Gödel, Bernays, and Hilbert

The correspondence between Paul Bernays and Kurt Gödel is one of the most extensive in the two volumes of Gödel's Collected Works (1986–2003) devoted to his letters of (primarily) scientific, philosophical, and historical interest. Ranging from 1930 to 1975, except for one long break, this correspondence engages a rich body of logical and philosophical issues, including the incompleteness theorems, finitism, constructivity, set theory, the philosophy of mathematics, and post-Kantian philosophy. In addition, Gödel's side of the exchange includes his thoughts on many topics that are not expressed elsewhere and testify to the lifelong warm, personal relationship that he shared with Bernays. I have given a detailed synopsis of the Bernays-Gödel correspondence, with explanatory background, in my introductory note to it in CW IV (pp. 41–79). My purpose here is to focus on only one group of interrelated topics from these exchanges, namely, the light that this correspondence – together with assorted published and unpublished articles and lectures by Gödel – throws on his perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert's program. In that connection, this piece has an important subtext, namely, the shadow of Hilbert that loomed over Gödel from the beginning to the end of his career.

Let me explain. Hilbert and Ackermann (1928) posed the fundamental problem of the completeness of the first-order predicate calculus in their logic text; Gödel (1929) settled that question in the affirmative in his dissertation a year later.